--- title: "Characteristics of Butterworth approximation" subject: "Filter Design" chapter: 2 key: "semester_5/filter_design:butterworth-characteristics" source: https://www.bctengineer.com/q/semester_5/filter_design/butterworth-characteristics site: BCT Engineering --- # Characteristics of Butterworth approximation Filter Design · Chapter 2: Approximation Methods - Asked in: 2082 Chaitra (new course), 3 marks - Video: [Filter Approximations: Overview of Butterworth, Chebyshev, Elliptic and Bessel Filter Approximation](https://www.youtube.com/watch?v=JXcbbBMrLYE) — ALL ABOUT ELECTRONICS - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/filter_design/butterworth-characteristics - Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-2 ## Question What are the characteristics of Butterworth low pass approximation? ## Short answer The Butterworth response is maximally flat at ω=0, monotonic with no ripple, and has magnitude |T(jω)|^2=1/(1+ε^2 ω^(2n)). At ω=1 (ε=1) it is -3 dB. Roll-off is 20n dB/decade. All-pole, poles lie on a circle in the left half-plane, phase is fairly linear. ## Answer The **Butterworth (maximally flat)** low-pass response has magnitude $$ |T(j\omega)|^2 = \frac{1}{1 + \varepsilon^2\omega^{2n}} $$ (normalized so that $\omega_p$ = 1 rad/s; with $\varepsilon = 1$ the half-power frequency is 1 rad/s). Its characteristics are: - **Maximally flat at $\omega = 0$:** the first $2n-1$ derivatives of $|T(j\omega)|^2$ are zero at $\omega = 0$, so the passband is as flat as possible. - **Monotonic:** no ripple in the passband or the stopband; the gain falls continuously with frequency. - $|T(j0)| = 1$; at $\omega = 1$ (when $\varepsilon = 1$) $|T| = 1/\sqrt{2}$ (−3 dB) for every order. - **Roll-off** of $20n$ dB/decade far into the stopband; the transition is wider than Chebyshev or elliptic for the same $n$. - **All-pole** function: all transmission zeros are at $\omega = \infty$. - **Poles lie on a circle** of radius $\omega_o = \varepsilon^{-1/n}$ in the left-half s-plane, equally spaced by $\pi/n$, symmetric about the real axis. - Phase is fairly linear (better than Chebyshev), with a moderate delay peak near the cut-off. - As $n \to \infty$ the response approaches the ideal brick-wall. ```text |T| 1 |----.__ | `. n = 2 0.707 - - - - :\. n = 4 | : \ \. | : `. `-._ +---------+---------`----> w 1 ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is true of the Butterworth magnitude response in the passband and stopband? A. Ripple in the stopband only B. Ripple in the passband only C. No ripple in either band; gain falls monotonically D. Equal ripple in both bands Answer: C. Butterworth is monotonic with no ripple anywhere; Chebyshev type I has passband ripple. (Key point: Monotonic response) 2. What does 'maximally flat at w = 0' mean? A. Poles are at w = 0 B. The gain is zero at w = 0 C. The first 2n-1 derivatives of |T|^2 are zero at w = 0 D. Gain is constant for all w Answer: C. The passband is as flat as possible near w = 0 because the first 2n-1 derivatives vanish. (Key point: Maximally flat) 3. Which statement about Butterworth poles and zeros is correct? A. Poles on a circle in the right-half plane B. All-pole: zeros at infinity; poles on a circle in the left-half plane C. Zeros on the jw axis and poles on an ellipse D. Poles on the jw axis Answer: B. The Butterworth function is all-pole with transmission zeros at infinity; poles lie on a circle, equally spaced, in the left half-plane. 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